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[MUSIC]. 

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Welcome back to Calculus One, and Welcome 
to Week 13 of our time together. 

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Well, we're almost to the end of the 
course. 

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Soon we will be posting the final exam, 
which will give you a few weeks to work 

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on that. 
But I wanted to say a little bit about 

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where we are. 
Last week we looked at how 

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anti-differentiation gives us the 
fundamental theorem of calculus, which 

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lets us evaluate integrals way more 
easily than by going back to the 

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definition in terms of Riemann sums. 
This week we continue on that quest, 

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we're going to look at u-substitution 
which is one really great technique for 

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actually finding some complicated 
antiderivatives. 

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Now, in light of this, I also want to say 
a little about how the course as a whole 

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has been organized. 
Remember way back at the beginning of the 

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course, we gave the definition of limit 
in terms of epsilons and deltas, but very 

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quickly, we abandoned those epsilons and 
deltas, and we gave a bunch of techniques 

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for evaluating limits. 
Things like the the limit of a sum is the 

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sum of the limits that the limits exist. 
And then, once we had easier techniques 

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for playing around with limits, we get 
some applications of limits. 

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A big one, being the limit of a 
difference quotient, which was the 

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definition of the derivative. 
But we didn't usually calculate the 

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derivative by going back to that 
definition. 

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Usually we used our techniques of 
differentiation like, the derivative is 

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of a sum is the sum of derivatives to 
make calculating derivatives way easier. 

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And then we did some applications of 
differentiations, right, things like 

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optimization problems, related rates 
problems, and we then did definition of 

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integration, right, in terms of Riemann 
sums. 

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But those Riemann sums were way to 
complicated, so we very quickly abandoned 

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Riemann sums for a, a, better technique 
Involving the fundamental theorem of 

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calculus. 
In this week, with use substitution, we 

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learned another technique for finding 
anti-derivatives, that is another 

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technique for evaluating definite 
integrals. 

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And in the last couple weeks, we're 
going to look at some more techniques and 

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some applications of integration. 
There's not much time left, hang in 

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there. 
We're all going to make it to the end 

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together.